A111647 -id:A111647

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A111648

a(n) = A001541(n)^2 + A001653(n+1)^2 + A002315(n)^2.

+0
3

3, 83, 2811, 95483, 3243603, 110187011, 3743114763, 127155714923, 4319551192611, 146737584833843, 4984758333158043, 169335045742539611, 5752406796913188723, 195412496049305876963

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,1

LINKS

Ray Chandler, Table of n, a(n) for n = 0..652

Index entries for linear recurrences with constant coefficients, signature (35,-35,1).

FORMULA

a(n) = A038761(n)^2 + 2, e.g., 95483 = 309^2 + 2.

a(n) = A001652(2*n+1) - A001109(n+1)^2 - Sum_{k=1..n-1} A038723(2*n), e.g., 95483 = 137903 - 204^2 - (23 + 781).

For n > 0, 2*a(n) + A001652(2*n-1) = A001653(2*n+2), e.g., 2*2811 + 119 = 5741.

G.f.: -(11*x^2-22*x+3) / ((x-1)*(x^2-34*x+1)). - Colin Barker, Dec 14 2014 (Empirical g.f. confirmed for more terms and recurrence of source sequences. - Ray Chandler, Feb 05 2024)

EXAMPLE

a(1) = 83 = 3^2+5^2+7^2.

MATHEMATICA

LinearRecurrence[{35, -35, 1}, {3, 83, 2811}, 20] (* Paolo Xausa, Feb 06 2024 *)

CROSSREFS

Cf. A001541, A001653, A002315, A111647, A111649.

KEYWORD

nonn,easy

AUTHOR

Charlie Marion, Aug 24 2005

STATUS

approved

A111649

a(n) = A001541(n)*A001653(n+1) + A001541(n)*A002315(n) + A001653(n+1)*A002315(n).

+0
3

3, 71, 2379, 80783, 2744211, 93222359, 3166815963, 107578520351, 3654502875939, 124145519261543, 4217293152016491, 143263821649299119, 4866752642924153523, 165326326037771920631, 5616228332641321147899

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,1

LINKS

Ray Chandler, Table of n, a(n) for n = 0..652

Index entries for linear recurrences with constant coefficients, signature (35,-35,1).

FORMULA

a(n) = A001653(2n+2) - 2*A002315(n)^2, e.g., 2379 = 5741 - 2*41^2;

a(n) = A001652(2n) + A002315(n)^2 + 2, e.g., 2379 = 696 + 41^2 + 2;

a(n) = 2*A046176(n+1)+1, e.g., 2379 = 2*1189 + 1.

G.f.: (x^2+34*x-3) / ((x-1)*(x^2-34*x+1)). - Colin Barker, Dec 14 2014 [adjusted for corrected term and empirical g.f. confirmed for more terms and recurrence of source sequences. - Ray Chandler, Feb 05 2024]

EXAMPLE

a(1) = 71 = 3*5 + 3*7 + 5*7.

MATHEMATICA

LinearRecurrence[{35, -35, 1}, {3, 71, 2379}, 20] (* Paolo Xausa, Feb 06 2024 *)

CROSSREFS

Cf. A001541, A001653, A002315, A111647, A111648.

KEYWORD

nonn

AUTHOR

Charlie Marion, Aug 24 2005

EXTENSIONS

a(3) = 80783 corrected by Ray Chandler, Feb 05 2024

STATUS

approved

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